A characterization of non-dunford-pettis operators onL 1

It is shown that ifT :L ¹ →L ¹ is a bounded linear operator failing the Dunford-Pettis property, thenT fixes a copy of ⊕_ll¹ (l ²).     

Dunford-pettis operators onL 1 and the Radon-Nikodym property

Using the duality between Dunford-Pettis operators onL ¹ and Pettis-Cauchy martingales, we prove that the Dunford-Pettis operators fromL ¹ intoL ¹ form a lattice. We show also that a Banach spaceX has the Radon-Nikodym property provided the Dunford-Pettis members of ℒ(L ¹,X) are representable. The lifting of dual valued Dunford-Pettis operators is investigated. Some remarks are included.     

On the halo conjecture and maximal convolution operators for locally compact groups

Acta Mathematica Hungarica - The halo conjecture in the theory of differentiation of integrals states that if the maximal operator MB corresponding to a differentiation basis B is of restricted...     

A continuous linear right inverse of the representation operator and applications to the convolution operators

Rostov-na-Donu. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 1, pp. 70–84, January–February, 1993.     

Simultaneous similarity of convolution operators to powers of the operator of integration

A necessary and sufficient condition is derived for a pair of convolution integral operators to be simultaneously similar to powers of the operator of integration. Also, simultaneous similarity of a pair of general integral operators to a pair of integral operators of convolution type is discussed. The results are compared with the matrix case.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1432–1437, October, 1995.The author would like to thank H. Bart and M. M. Malamud for their helpful comments and suggestions.     

A convolution operator in weighted spaces

Convolution operators in R ^m with a homogeneous function of an arbitrary complex order are considered in spaces with weighted norms. The role of the weight is played by some power of the distance to a k-dimensional plane. The fundamental content of the paper consists in the investigation of the transversal operators (TO), which arise from convolutions as a result of a Fourier transform along the mentioned plane and a change of variables. Formulas for the composition of TO are derived and the (finite-dimensional) kernels and cokernels of elliptic TO are described. Problems for TO are indicated, containing additional boundary conditions or potentials. The properties of the convolution operators are obtained from the corresponding properties of the TO.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 208–237, 1990.     

A new approach to the convolution operator on a finite interval

The convolution operator on a finite interval defined on a space ofL ₂ functions is studied by relating it to a singular integral operator acting on a space of functions defined on a system of two parallel straight lines in the complex plane . The approach followed in the paper applies both to the case where the Fourier transform of the kernel functions is anL function and to the case where the kernel function is periodic, thus yielding a unified treatment of these two classes of kernel functions. In the non-periodic case it is possible, for a special class of kernel functions, to study the invertibility property of the operator giving an explicit formula for the inverse. An example is presented and generalizations are suggested.     

Weak type (1, 1) inequalities of maximal convolution operators

In this paper we study the behavior of the constants which appear in the weak type (1, 1) inequalities for maximal convolution operators by means of discrete methods. One of the first applications of these techniques will give us a very simple proof of the ergodic theorem. We also present partial results in order to investigate the best constant in the weak type (1, 1) inequality for the Hardy-Littlewood centered maximal operator in dimension one. In dimension bigger than one we also obtain some lower bounds for that constant.     

Wiener-Hopf operators with oscillating symbols and convolution operators on a union of intervals

The present work was sponsored by JNICT (Portugal) under grant n. 87422/MATM and Programa Ciência.     

A remark on the behaviour ofL p-multipliers and the range of operators acting onL p-spaces

In this paper, new results are obtained concerning the uniform approximation property (UAP) inL ^p-spaces (p≠2,1,∞). First, it is shown that the “uniform approximation function” does not allow a polynomial estimate. This fact is rather surprising since it disproves the analogy between UAP-features and the presence of “large” euclidian subspaces in the space and its dual. The examples are translation invariant spaces on the Cantor group and this extra structure permits one to replace the problem with statements about the nonexistence of certain multipliers in harmonic analysis. Secondly, it is proved that the UAP-function has an exponential upper estimate (this was known forp=1, ∞). The argument uses Schauder’s fix point theorem. Its precise behaviour is left unclarified here. It appears as a difficult question, even in the translation invariant context.     

Functional calculi for convolution operators on a discrete, periodic, solvable group

Suppose T is a bounded self-adjoint operator on the Hilbert space L²(X,μ) and let